[Paper Review] Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations

Sander Rhebergen, Garth N. Wells|arXiv (Cornell University)|Jan 15, 2018

Advanced Numerical Methods in Computational Mathematics26 references44 citations

TL;DR

This paper presents optimal preconditioners for a hybridized discontinuous Galerkin (HDG) finite element method applied to the Stokes equations, leveraging static condensation to reduce the global system size. By proving spectral equivalence between the pressure Schur complement and a mass matrix, the authors develop block-diagonal and block symmetric Gauss–Seidel preconditioners that yield iteration counts independent of mesh size, ensuring scalability for large-scale problems in 2D and 3D.

ABSTRACT

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners.

Motivation & Objective

• To develop efficient iterative solvers for the statically condensed linear system arising from a hybridized discontinuous Galerkin (HDG) discretization of the Stokes equations.
• To ensure scalability of the solution process by constructing preconditioners whose performance is independent of mesh size.
• To preserve the key properties of the HDG method—local mass conservation, pointwise divergence-free velocity, and energy stability—while enabling efficient solution of the condensed system.
• To analyze and numerically validate preconditioners that maintain optimal convergence rates across different polynomial degrees and mesh types (simplicial, quadrilateral, hexahedral).

Proposed method

• Apply static condensation to eliminate cell-wise velocity degrees of freedom, reducing the global problem size while preserving the pressure Schur complement.
• Prove spectral equivalence between the pressure Schur complement and a discrete mass matrix using boundedness and inf-sup stability results from prior analysis.
• Construct two block-diagonal preconditioners: one based on the spectrally equivalent matrix C (derived from the Schur complement), and one based on the pressure mass matrix M.
• Develop block symmetric Gauss–Seidel preconditioners using the system matrix structure and approximate inverses via algebraic multigrid (AMG).
• Use AMG with V-cycles to approximate inverse operators (e.g., for ¯R−1, M−1, C−1), with Gauss–Seidel smoothers applied in pre- and post-smoothing steps.
• Implement the method in MFEM with PETSc solvers and BoomerAMG for algebraic multigrid, enabling numerical testing on unstructured and structured meshes.

Experimental results

Research questions

• RQ1 Can optimal preconditioners be constructed for the statically condensed HDG system of the Stokes equations, ensuring convergence independent of mesh size?
• RQ2 Does the Schur complement of the condensed system remain spectrally equivalent to a mass matrix, enabling efficient preconditioning?
• RQ3 How do block-diagonal and block symmetric Gauss–Seidel preconditioners compare in performance across different mesh types and polynomial degrees?
• RQ4 Does right-preconditioning with GMRES outperform left-preconditioning with MINRES in terms of iteration count for the same preconditioner?
• RQ5 Can the proposed preconditioners maintain robustness for high-order polynomial approximations and complex geometries (e.g., lid-driven cavity flow) in 2D and 3D?

Key findings

• The Schur complement of the statically condensed HDG system is spectrally equivalent to a discrete pressure mass matrix, enabling the use of established preconditioning theory.
• The block-diagonal preconditioner based on the spectrally equivalent matrix C achieves mesh-independent convergence, with iteration counts stable across refinement levels in both 2D and 3D.
• For 3D simplicial meshes with 1,789,952 DOFs, the ¯PC preconditioner required only 96 MINRES iterations to reach a 10−8 residual tolerance.
• Right-preconditioned GMRES reduced iterations from 153 (left-preconditioned MINRES) to 37 for the same 3D simplicial grid, demonstrating significant performance gains.
• The block symmetric Gauss–Seidel preconditioner ¯PSGS_C achieved 102 iterations in 2D and 153 in 3D for the largest simplicial mesh, showing robustness across problem sizes.
• The preconditioner based on M (pressure mass matrix) underperformed relative to C, while the symmetric Gauss–Seidel variant showed no advantage using C over M.